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A remark on Tsirelson's stochastic differential equation

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A remark on Tsirelson's stochastic differential equation SÉMINAIRE DE PROBABILITÉS (STRASBOURG) MICHEL ÉMERY WALTER SCHACHERMAYER A remark on Tsirelson’s stochastic differential equation Séminaire de probabilités (Strasbourg), tome 33 (1999), p. 291-303. <http://www.numdam.org/item?id=SPS_1999__33__291_0> © Springer-Verlag, Berlin Heidelberg New York, 1999, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml), implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ A REMARK ON TSIRELSON’S STOCHASTIC DIFFERENTIAL EQUATION M. Émery and W. Schachermayer ABSTRACT. 2014 Tsirelson’s stochastic differential equation is called "celebrated and mysteri- ous" by Rogers and Williams [16]. This note aims at making it a little more celebrated and a little less mysterious. ’ Using a deterministic time-change, we translate the study of Tsirelson’s equation into the study of "eternal" Brownian motion on the circle. This allows us to show that the filtration generated by any solution of Tsirelson’s equation is also generated by some Brownian motion (which, however, cannot be the Brownian motion driving the equation, because the equation has no strong solution). Introduction The so-called innovation problem is a remarkable phenomenon in the theory of filtered probability spaces; see for instance § 5.4 of von Weizsacker ~24~ . When the answer to the innovation problem is negative, some kind of creation of information occurs. This may happen in discrete, or continuous time (by discrete time, we refer to processes parametrized by Z). In discrete time

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